Lecture 1 EVOLUTIONARY GAME THEORY Toulouse School of Economics

Transcription

1 Lecture 1 EVOLUTIONARY GAME THEORY Toulouse School of Economics Jörgen Weibull November 13, 2012

2 1 Economic theory and as if rationality The rationalistic paradigm in economics: Savage rationality [Leonard Savage: The Foundations of Statistics, 1954] Each economic agent s behavior derived from maximization of some goal function (utility, profit), under given constraints and information The as if defence by Milton Friedman (1953): The methodology of positive economics Firms that do not take profit-maximizing actions are selected against in the market But is this claim right? Under perfect competition? Under imperfect competition?

3 Evolutionary theorizing: Smith De Mandeville, Malthus, Darwin, Maynard Darwin: exogenous environment - perfect competition Maynard Smith: endogenous environment - imperfect competition

4 2 Game theory A mathematically formalized theory of strategic interaction. Applications abound, in economics in political science in biology in computer science

5 3 John Nash s two interpretations of equilibrium Nash (Ph.D. thesis, Mathematics Department, Princeton, 1950): 1. The rationalistic (or epistemic) interpretation 2. The mass action (or evolutionary) interpretation

6 3.1 The rationalistic interpretation 1. The players interact exactly once (true even if the interaction itself is a repeated game). 2. The players are rational in the sense of Savage (1954) 3. Each player knows the game in question (strategy sets, payoffs to all players) However, this clearly does not imply that they will play a Nash equilibrium. Hence, assume in addition that 4. Thegameandallplayers rationalityiscommon knowledge

7 Example 3.1 (Coordination game) Example 3.2 (Matching Pennies)

8 Example 3.3 A game with a unique Nash equilibrium. This equilibrium is strict; any deviation is costly

9 Conclusion The rationalistic interpretation, even under CK of the game and rationality, does not in general imply equilibrium play [In fact, this combined hypothesis implies rationalizability.]

10 3.2 Nash s mass-action interpretation 1. For each player role in the game: a large population of identical individuals 2. The game is recurrently played, at times = (or at Poisson arrival times) by randomly drawn individuals, one from each player population 3. Individuals play pure strategies 4. Individuals observe empirical samples of earlier behaviors and outcomes and avoid suboptimal actions [According to Nash (1950), they need not know the strategy sets or payoff functions of other player roles. They do not even have to know they play a game.]

11 A population-statistical distribution over the pure strategies in a player role constitute a mixed strategy Nash s (informal) claim: If all individuals avoid suboptimal pure strategies, and the population distribution is stationary ( stable ), then it constitutes a Nash equilibrium Statistical independence across player populations joint probability distribution over pure-strategy profiles constitutes a mixedstrategy profile Zero probability for suboptimal pure strategies each playerpopulation s mixed strategy is a best reply Reconsider the above examples in this (informal) interpretation

12 Conclusion The mass-action interpretation does not in general imply equilibrium play, but it almost does it, and holds promise In fact, evolutionary game theory provides methods and concepts to rigorously explore these promises The folk theorem of evolutionary game theory: If the population process converges from an interior initial state, then the limit distribution is a Nash equilibrium If a stationary population distribution is stable, then it constitutes a Nash equilibrium

13 If the population process is convex-monotone, then non-rationalizable strategies have zero asymptotic probability, in all finite two-player games

14 4 Evolutionary game theory Evolutionary process = = mutation process + selection process The unit of selection: usually strategies ( strategy evolution ), sometimes utility functions ( preference evolution ) 1. Evolutionary stability: focus on robustness to mutations 2. Replicator dynamic: focus on selection. [Robustness to mutations by way of dynamic stability] 3. Stochastic stability: both selection and mutations

15 5 Evolutionary stability under strategy evolution ESS = evolutionarily stable strategy [Maynard Smith and Price (1972), Maynard Smith (1973)] - a strategy that cannot be overturned, once it has become the convention in a population

16 Heuristically 1. A large population of individuals who are recurrently and (uniformly) randomly matched in pairs to play a finite and symmetric two-player game 2. Initially, all individuals always use the same pure or mixed strategy,, the incumbent (or resident) strategy 3. Suddenly, a small population share 0switchtoanotherpureor mixed strategy,, themutant strategy 4. If the incumbents (residents) on average do better than the mutants, then is evolutionarily stable against

17 5. is called evolutionarily stable if it is evolutionarily stable against all mutations 6=

18 Domain of analysis Symmetric finite two-player games in normal form Definition 5.1 A finite and symmetric two-player game is any normal-form game =( ) with = {1 2}, 1 = 2 = = {1 } and 2 ( ) = 1 ( ) for all. Payoff bimatrix ( ), where =( ), =( ) Game symmetric iff =.

19 Example 5.1 (Prisoners dilemma) Payoff bimatrix: = Ã ! = Ã ! Symmetric since =.

20 Write for ( ), the mixed-strategy simplex: = { R + : X =1} Write the payoff to any strategy, whenusedagainstanystrategy as ( ) = Note that the first argument is own strategy, second argument the other party s strategy. While the prisoner dilemma is symmetric, matching-pennies is not

21 Example 5.2 (Matching Pennies) Payoff bimatrix: = Ã ! = Ã ! Here 6=. Not a symmetric game. Thus, matching pennies games fall outside the domain of evolutionary stability analysis. However, if player roles are randomly assigned, with equal probability for both role-allocations, then the so-defined metagame is symmetric, and evolutionary stability analysis applies to the metagame.

22 Example 5.3 (Coordination game) Payoff bimatrix: = = Ã ! A doubly symmetric game: = = (thisisanexampleofa potential game, Monderer and Shapley, 1996)

23 Best replies to : ( ) ={ : ( ) ³ 0 0 } This defines a correspondence from to itself: : [This is distinct from the usual BR correspondence, which maps 2 to ] Let = { : ( )} Note ( ) is a symmetric Nash equilibrium Proposition 5.1 6=. Proof: Application of Kakutani s Fixed-Point Theorem.

24 5.1 Definition of ESS We are now in a position to define evolutionary stability exactly: Definition 5.2 is an evolutionarily stable strategy (ESS) if for every strategy 6= (0 1) such that for all (0 ): [ +(1 ) ] [ +(1 ) ] Post-entry mixture : = +(1 ) a convex combination of and, a point on the straight line between them.

25 Let denote the set of ESSs Note that an ESS has to be a best reply to itself: if then ( ) ( ) forall Hence Note also that an ESS has to be a better replytoitsalternativebest replies than they are to themselves: if, ( ) and 6=, then ( ) ( )

26 Proposition 5.2 if and only if for all 6= : ( ) ( ) and ( ) = ( ) ( ) ( )

27 5.2 Examples Prisoner s dilemma = = { }

28 5.2.2 Coordination game = ½ ¾ 3 = { } The mixed NE is perfect and even proper, but not evolutionarily stable

29 5.2.3 Partnership game Small start-up businesses with two partners, or pairs of students assigned to write an essay together Each partner has to choose between work (or contribute ) and shirk (or free-ride ) - If both choose W: good outcome for both - If one chooses W and the other S: loss to the first but gain to the second - If both choose S: heavy loss to both Example of payoff bimatrix:

30 Symmetric game, but not a Prisoners Dilemma: S does not dominate W: it is better to work if the other shirks Consider a large pool of individuals and random matching What do you think would happen? A tendency to work? To shirk? Anytendencyatall? Any ESS? This is strategically equivalent to a so-called hawk-dove game, where work = dove and shirk = hawk

31 1. Unique symmetric NE: randomize uniformly, =( ), = { }. Hence { } 2. an ESS iff ( ) ( ) 6= 3. Equivalently: or 1 2 [ (1 1 )] (1 1 ) (1 1 ) or 4 µ

32 4. True, hence is an ESS! Graphical illustration of payoff difference ( ) ( ): payoff difference y1

33 Some games have no ESS. For instance, when all payoffs arethesame. But also in more interesting games such as Example 5.4 (Rock-scissors-paper) Rock beats Scissors, Scissors beat Paper, and Paper beats Rock: = Unique Nash equilibrium: = ³ All pure strategies are best replies and do as well against themselves as does against them. =.

34 THE END Literature: Chapter 9 in van Damme (1991) or Chapter 2 in Weibull (1995).